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When mathematics teachers consider acting on behalf of the discipline, what assumptions do they make?

Abstract

We discuss findings from a study of 226 K-12 mathematics teachers who rated the appropriateness of instructional actions in a scenario-based assessment. The teaching actions shown in the scenarios were designed to represent mathematics teachers fulfilling their obligation to the discipline of mathematics. Our analysis of the participants' follow-up responses provides insight into how other professional obligations interact with teachers' obligation to the discipline. We describe our method for detecting what we call conditional construals—assumptions participants regarded as critical for rating the appropriateness of instructional actions. In addition, we report preliminary findings about how those assumptions can be accounted for in terms of the professional obligations of mathematics teaching.
WHEN MATHEMATICS TEACHERS CONSIDER ACTING ON BEHALF OF THE
DISCIPLINE, WHAT ASSUMPTIONS DO THEY MAKE?
!!!Amanda!Milewski! ! !!Ander!Erickson!!!!!!!!!!!!!!!!!Patricio!Herbst!!!!!!!!!!!!!!!!!Justin!Dimmel!
University!of!Michigan!!!!!!!!!University!of!Michigan!!!!!!!University!of!Michigan!!!!University!of!Maine!
!
We discuss findings from a study of 226 K-12 mathematics teachers who rated the appropriateness
of instructional actions in a scenario-based assessment. The teaching actions shown in the
scenarios were designed to represent mathematics teachers fulfilling their obligation to the
discipline of mathematics. Our analysis of the participants’ follow-up responses provides insight
into how other professional obligations interact with teachers’ obligation to the discipline. We
describe our method for detecting what we call conditional construals—assumptions participants
regarded as critical for rating the appropriateness of instructional actions. In addition, we report
preliminary findings about how those assumptions can be accounted for in terms of the professional
obligations of mathematics teaching.
Keywords: Instructional Activities and Practices, Teacher Knowledge, Teacher Beliefs, Research
Methods
Introduction
Scenario-based assessments are widely used to assess judgment in professional fields, such
as medicine, police work, army tactics, and human resources (Weekley & Polyhardt, 2006). In this
study, we analyzed a set of open-ended responses to scenario-based assessment items of K-12
mathematics teaching. Open responses from participating teachers followed forced-choice ratings
of instructional actions shown within scenarios of K-12 mathematics classrooms. There were two
parts to the analysis. The first part of the analysis involved validating a procedure for identifying
when a response contained linguistic markers that indicated that the appropriateness of an
instructional action was dependent on an element of the scenario (e.g., traits of the students, the
amount of available time, the content that had been previously covered) that was unspecified in the
representation that they viewed. We call such markers conditional construals. For the first part of
the analysis, we analyzed responses from 226 teachers to 15 items (n=3405) designed to probe
teachers’ professional obligation to the discipline of mathematics (Herbst & Chazan, 2012). The
second part of the analysis was a qualitative examination of the kinds of additional information
participants stated they would need to know about a scenario in order to rate the appropriateness of
an instructional action. The qualitative analysis examined the responses to 2 of the 15 items that had
the highest incidence of responses that contained a conditional construal. In this paper we report on
the results of the preliminary qualitative analysis of the responses to these 2 high construal items.
We looked for additional aspects of the represented scenario that teachers considered critical
in deciding whether an action was appropriate. The following research questions guided our
inquiry: 1) What types of assumptions do teachers make in order to justify instructional decisions
that respond to a professional obligation to the discipline? 2) How do the different professional
obligations support or constrain teachers’ departures from “business as usual” in favor of their
obligation to the discipline of mathematics? Our purpose with this study is to investigate the types
of assumptions teachers make when considering instructional actions that respond to the
professional obligation to the discipline of mathematics.
Theoretical Framework for Research
Herbst and Chazan’s (2012) account of the practical rationality of mathematics teaching
attempts to synthesize rational actor theories of teacher decision making (e.g., Calderhead, 1996;
Schoenfeld, 2010) with accounts of teaching as a sociocultural activity (Stigler & Hiebert, 1999).
Herbst and Chazan (2012) propose that while teachers have personal resources such as knowledge
and beliefs, teachers also play roles in activity systems (e.g., algebra instruction) that have
customary norms (Much & Shweder, 1978). Within such activity systems, teachers are accountable
to four professional obligations—to the discipline of mathematics, to individual students, to the
classroom community, and to the institutions of schooling (Herbst & Chazan, 2012). While actions
following customary norms may be enacted without reflection, a teacher’s deviations from those
norms, whether motivated by individual resources or in response to social demands, need
justification (Buchmann, 1986). Herbst and Chazan (2012) argue that the four obligations provide a
basis for the sources of professional justification. For example, while the obligation to the discipline
of mathematics would require a teacher to limit the range of symbolic strings that she allows herself
to call equations (e.g., 2x = 5 is an equation but 3x -7 is not an equation), this same obligation could
also justify her calling equation a symbolic string like 2x = x2 in spite of the fact that the beginner’s
algebra curriculum might not include it under the equations that algebra students learn to solve
(Herbst & Chazan, 2012).
While much work has been carried out to establish links between teachers’ beliefs and
conceptions about mathematics and instructional practices (Skott, 2001; Stipek et al., 2001), the
notion of practical rationality suggests that teachers’ decision-making is not driven by individual
resources alone. Rather, teachers’ decisions are context-dependent and a fuller understanding of the
resources that they can employ for departing from normative practice can be achieved through the
use of instruments that represent the situated nature of practice. As such, we approach the work of
understanding teachers’ decision making by presenting them with scenarios of instruction in which
the represented teacher departs from instructional norms in order to act on behalf of the professional
obligation to the discipline.
Methodology
We conducted our study on data collected from K-12 inservice mathematics teachers with a
15 item scenario-based instrument designed to measure recognition of teachers’ professional
obligation to the discipline of mathematics. Within each item, participants examined a scenario in
which a teacher departs from an instructional norm in order to attend to one of the professional
obligations. Participants were asked to rate the extent of their agreement (using a 6-point Likert-like
response format from strongly disagree to strongly agree) with the teacher’s action. Participants
were then prompted to comment on their ratings. In order to identify conditional construals in these
responses, we located circumstances of contingency (Halliday & Matthiessen, 2004, p. 271) which
are accompanied by linguistic markers such as “depending on”, “as long as”, and “assuming”. In
particular, we examined the nominal group introduced by the circumstances of contingency. These
nominal groups are usually “a noun denoting an entity whose existence is conditional, a noun
denoting an event that might eventuate, or a nominalization denoting a reified process or quality”
(Halliday & Matthiessen, 2004, p. 272). We present an example with the following response, “This
depends on timing. I think it can be valuable information for students to see common errors played
out in a solution, but time will dictate whether or not that can happen in any given scenario.” We
argue that “timing” is marked here as a circumstance of contingency by the prepositional phrase
“this depends on”. We then categorized those circumstances by professional obligation. In this
example, we placed the conditional construal in the institutional category because the participant
indicates that the appropriateness of the action is contingent on the amount of time available, and
time constraints are imposed by the institution.
Findings
We found that teachers consider the other obligations—to the students as individuals, to the
classroom community, and to the institution—as they consider whether to respond to their
professional obligation to the discipline of mathematics. We illustrate these categories of
conditional construals by sharing teachers’ responses from the two items with the highest number of
responses that contained conditional construals. The analysis reported below qualitatively assesses
the content of these conditional construals, with respect to the professional obligations.
In the first item, participants view a scenario in which the teacher announces that students
seem to be on the right track and, rather than spend time on practice problems (which we
hypothesize would be the normative action in such a situation), the teacher decides to discuss
mathematical theory related to the topic at hand. Participants were asked to rate the statement: “The
teacher should give students additional practice problems, rather than elaborate on mathematical
theory.” Of the 226 participant responses, 16% indicated that their decision depended on
assumptions they had made about the scenario. In the second item, participants viewed a scenario in
which a student makes a mistake while working a problem at the board; rather than point out the
mistake directly (which we hypothesize would be the normative action in such a situation), the
teacher allows the student to build on the mistake. Such an action could be justifiable on the
disciplinary grounds that the teacher is creating an opportunity for a contradiction or anomaly to
arise from mathematical reasoning that builds on the mistake. Participants were asked to rate the
statement: “The teacher should correct the mistake observed, rather than ask students to build on
mistaken work,” and we found that 7% of the participants indicated that their decision depended on
assumptions they made about the scenario.
These items highlight the variety of factors that teachers consider in order to make
instructional decisions that respond to the disciplinary obligation and demonstrate how the role of
the other professional obligations can vary depending on the scenario. For example, while there
were 30 participants who referenced the individual obligation in their responses to the first item, the
individual obligation was used in different ways. One perspective suggested that knowing the
“level” of the class is essential for determining whether or not a particular instructional action is
appropriate; another perspective suggested that it was of primary importance to determine whether
students had been taught relevant information. While these are both manifestations of the obligation
to the individual student, they have different implications for how the teacher perceives the agency
of the students and the responsibility of the teacher. Such differences manifested themselves across
items as well: In the case of the first item, there were 30 references to the individual obligation, but
there were only 9 references to the individual obligation for the second item. Ongoing analysis is
being to conducted to perceive patterns in the way each of the obligations are manifested within
participants' construals.
Study’s Significance
Mathematics instruction is beholden to the discipline of mathematics as the source of
legitimacy of the knowledge at stake in the classroom as well as the practices that govern the
development of that knowledge. Nonetheless, there is often a tension between the type of
instruction advocated by mathematicians and mathematics educators and the everyday reality of K-
12 education (Schoenfeld, 2004). We have demonstrated how we have been able to collect a corpus
of data from K-12 teachers of mathematics describing the circumstances that dictate whether they
approve of a fellow teacher departing from customary classroom practice in order to respond to
their perceived professional obligation to the discipline of mathematics. Our analysis of that corpus
suggests that, for these teachers, their professional obligation to the individual students (e.g.
students’ understanding), to the discipline of mathematics (e.g. mathematical value), to the
classroom community (e.g. class norms), and to the institution of school (e.g. time) all have a
bearing on their approval of such decisions. It remains an open question for us whether individual
differences account for part of the variance in participants’ conditional construals. To explore this,
we could examine how proxies for such differences, such as mathematical knowledge for teaching
and experience, are related to the ways that they see the professional obligations impinging on their
practice.
While surveys of teachers’ beliefs about mathematics provide one way of conceptualizing
how the discipline of mathematics may influence instructional decisions, our work presents a
different approach by suggesting that there are aspects of the professional position (for example, the
need to teach a given curriculum, a limited allowance of time, the characteristics of the class being
taught, the various characteristics and needs of individuals one is assigned to teach) that may
mediate a teachers’ willingness to respond to the discipline regardless of their beliefs about the
subject. As a better model is developed of the way that the professional obligations interact both
with one another and with personal resources of the mathematics teacher, and as the different
aspects of the professional obligations themselves are further analyzed as we have begun to do here,
we will be closer to being able to understand the hidden factors that obstruct or facilitate the take-up
of instructional practices that depart from the norm in order to align with the discipline of
mathematics.
References
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Calderhead, J. (1996). Teachers: Beliefs and knowledge. In D.C Berliner & R.C. Calfee (Eds.), Handbook of
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Halliday, M., Matthiessen, C. M., & Matthiessen, C. (2014). An introduction to functional grammar. Routledge.
Herbst, P., & Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics
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Much, N. C., & Shweder, R. A. (1978). Speaking of rules: The analysis of culture in breach. New Directions for Child
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Schoenfeld, A. H. (2010). How we think: A Theory of Goal-Oriented Decision Making and its Educational
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